Editing Quantum error correction (section)

==Experimental realization==
There have been several experimental realizations of CSS-based codes. The first demonstration was with [[Nuclear magnetic resonance quantum computer|nuclear magnetic resonance qubits]].<ref>{{cite journal | last1 = Cory | first1 = D. G. | last2 = Price | first2 = M. D. | last3 = Maas | first3 = W. | last4 = Knill | first4 = E. | last5 = Laflamme | first5 = R. | last6 = Zurek | first6 = W. H. | last7 = Havel | first7 = T. F. | last8 = Somaroo | first8 = S. S. | year = 1998| title = Experimental Quantum Error Correction | journal = Phys. Rev. Lett. | volume = 81 | issue = 10| pages = 2152–2155 | doi = 10.1103/PhysRevLett.81.2152 | arxiv = quant-ph/9802018 | bibcode = 1998PhRvL..81.2152C | s2cid = 11662810 }}</ref> Subsequently, demonstrations have been made with linear optics,<ref>{{cite journal | last1 = Pittman | first1 = T. B. | last2 = Jacobs | first2 = B. C. | last3 = Franson | first3 = J. D. | year = 2005 | title = Demonstration of quantum error correction using linear optics | journal = Phys. Rev. A | volume = 71 | issue = 5| page = 052332 | doi = 10.1103/PhysRevA.71.052332 | arxiv = quant-ph/0502042 | bibcode = 2005PhRvA..71e2332P | s2cid = 11679660 }}</ref> trapped ions,<ref>{{cite journal | last1 = Chiaverini | first1 = J. | last2 = Leibfried | first2 = D. | last3 = Schaetz | first3 = T. | last4 = Barrett | first4 = M. D. | last5 = Blakestad | first5 = R. B. | last6 = Britton | first6 = J. | last7 = Itano | first7 = W. M. | last8 = Jost | first8 = J. D. | last9 = Knill | first9 = E. | last10 = Langer | first10 = C. | last11 = Ozeri | first11 = R. | last12 = Wineland | first12 = D. J. | year = 2004 | title = Realization of quantum error correction | journal = Nature | volume = 432 | issue = 7017| pages = 602–605 | doi = 10.1038/nature03074 | pmid = 15577904 | bibcode = 2004Natur.432..602C | s2cid = 167898 }}</ref><ref>{{cite journal | last1 = Schindler | first1 = P. | last2 = Barreiro | first2 = J. T. | last3 = Monz | first3 = T. | last4 = Nebendahl | first4 = V. | last5 = Nigg | first5 = D. | last6 = Chwalla | first6 = M. | last7 = Hennrich | first7 = M. | last8 = Blatt | first8 = R. | year = 2011 | title = Experimental Repetitive Quantum Error Correction | journal = Science | volume = 332 | issue = 6033| pages = 1059–1061 | doi = 10.1126/science.1203329 | pmid = 21617070 | bibcode = 2011Sci...332.1059S | s2cid = 32268350 }}</ref> and superconducting ([[transmon]]) qubits.<ref>{{cite journal | last1 = Reed | first1 = M. D. | last2 = DiCarlo | first2 = L. | last3 = Nigg | first3 = S. E. | last4 = Sun | first4 = L. | last5 = Frunzio | first5 = L. | last6 = Girvin | first6 = S. M. | last7 = Schoelkopf | first7 = R. J. | year = 2012 | title = Realization of Three-Qubit Quantum Error Correction with Superconducting Circuits | journal = Nature | volume = 482 | issue = 7385| pages = 382–385 | doi = 10.1038/nature10786 | pmid = 22297844 | arxiv = 1109.4948 | bibcode = 2012Natur.482..382R | s2cid = 2610639 }}</ref>

In 2016 for the first time the lifetime of a quantum bit was prolonged by employing a QEC code.<ref name=":5">{{Cite journal |last1=Ofek |first1=Nissim |last2=Petrenko |first2=Andrei |last3=Heeres |first3=Reinier |last4=Reinhold |first4=Philip |last5=Leghtas |first5=Zaki |last6=Vlastakis |first6=Brian |last7=Liu |first7=Yehan |last8=Frunzio |first8=Luigi |last9=Girvin |first9=S. M. |last10=Jiang |first10=L. |last11=Mirrahimi |first11=Mazyar |date=August 2016 |title=Extending the lifetime of a quantum bit with error correction in superconducting circuits |journal=Nature |volume=536 |issue=7617 |pages=441–445 |doi=10.1038/nature18949 |pmid=27437573 |issn=0028-0836 |bibcode=2016Natur.536..441O |s2cid=594116}}</ref> The error-correction demonstration was performed on [[Cat state|Schrödinger-cat states]] encoded in a superconducting resonator, and employed a [[quantum controller]] capable of performing real-time feedback operations including read-out of the quantum information, its analysis, and the correction of its detected errors. The work demonstrated how the quantum-error-corrected system reaches the break-even point at which the lifetime of a logical qubit exceeds the lifetime of the underlying constituents of the system (the physical qubits).

Other error correcting codes have also been implemented, such as one aimed at correcting for photon loss, the dominant error source in photonic qubit schemes.<ref>{{cite journal |last1=Lassen |first1=M. |last2=Sabuncu |first2=M. |last3=Huck |first3=A. |last4=Niset |first4=J. |last5=Leuchs |first5=G. |last6=Cerf |first6=N. J. |last7=Andersen |first7= U. L. |year=2010 |title=Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code |journal=Nature Photonics |volume=4 |issue=10| page=700 |doi=10.1038/nphoton.2010.168 | arxiv = 1006.3941 |bibcode=2010NaPho...4..700L |s2cid=55090423}}</ref><ref>{{cite journal| last1=Guo| first1=Qihao| last2=Zhao| first2=Yuan-Yuan| last3=Grassl| first3=Markus| last4=Nie| first4=Xinfang| last5=Xiang| first5=Guo-Yong| last6=Xin| first6=Tao| last7=Yin| first7=Zhang-Qi| last8=Zeng| first8=Bei| author8-link=Bei Zeng| title=Testing a quantum error-correcting code on various platforms| journal=Science Bulletin| year=2021| volume=66| issue=1| pages=29–35| doi=10.1016/j.scib.2020.07.033| pmid=36654309 |arxiv=2001.07998| bibcode=2021SciBu..66...29G| s2cid=210861230}}</ref>

In 2021, an [[Controlled NOT gate|entangling gate]] between two logical qubits encoded in [[Toric code|topological quantum error-correction codes]] has first been realized using 10 ions in a [[Trapped ion quantum computer|trapped-ion quantum computer]].<ref>{{cite news |title=Error-protected quantum bits entangled for the first time |url=https://phys.org/news/2021-01-error-protected-quantum-bits-entangled.html |access-date=30 August 2021 |work=phys.org |date=13 January 2021 |language=en}}</ref><ref>{{cite journal |last1=Erhard |first1=Alexander |last2=Poulsen Nautrup |first2=Hendrik |last3=Meth |first3=Michael |last4=Postler |first4=Lukas |last5=Stricker |first5=Roman |last6=Stadler |first6=Martin |last7=Negnevitsky |first7=Vlad |last8=Ringbauer |first8=Martin |last9=Schindler |first9=Philipp |last10=Briegel |first10=Hans J. |last11=Blatt |first11=Rainer |last12=Friis |first12=Nicolai |last13=Monz |first13=Thomas |title=Entangling logical qubits with lattice surgery |journal=Nature |date=13 January 2021 |volume=589 |issue=7841 |pages=220–224 |doi= 10.1038/s41586-020-03079-6 |pmid=33442044 |s2cid=219401398 |arxiv=2006.03071 |bibcode=2021Natur.589..220E |language=en |issn=1476-4687}}</ref> 2021 also saw the first experimental demonstration of fault-tolerant Bacon-Shor code in a single logical qubit of a trapped-ion system, i.e. a demonstration for which the addition of error correction is able to suppress more errors than is introduced by the overhead required to implement the error correction as well as fault tolerant Steane code.<ref>{{Cite web |last=Bedford |first=Bailey |date=2021-10-04 |title=Foundational step shows quantum computers can be better than the sum of their parts |website=phys.org |url=https://phys.org/news/2021-10-foundational-quantum-sum.html |access-date=2021-10-05 |language=en}}</ref><ref>{{Cite journal| last1=Egan| first1=Laird| last2=Debroy| first2=Dripto M.| last3=Noel| first3=Crystal| last4=Risinger| first4=Andrew| last5=Zhu| first5=Daiwei| last6=Biswas| first6=Debopriyo| last7=Newman| first7=Michael| last8=Li| first8=Muyuan| last9=Brown| first9=Kenneth R.| last10=Cetina| first10=Marko| last11=Monroe| first11=Christopher|date=2021-10-04| title=Fault-tolerant control of an error-corrected qubit| journal=Nature| volume=598| issue=7880| pages=281–286| language=en| doi=10.1038/s41586-021-03928-y| pmid=34608286| bibcode=2021Natur.598..281E| s2cid=238357892| issn=0028-0836}}</ref><ref>{{Cite journal| last=Ball| first=Philip| date=2021-12-23| title=Real-Time Error Correction for Quantum Computing| journal=Physics| language=en| volume=14| at=184| s2cid=245442996| doi=10.1103/Physics.14.184| bibcode=2021PhyOJ..14..184B| doi-access=free}}</ref> In a different direction, using an encoding corresponding to the Jordan-Wigner mapped Majorana zero modes of a Kitaev chain, researchers were able to perform quantum teleportation of a logical qubit, where an improvement in fidelity from 71% to 85% was observed.<ref>{{Cite journal| last1=Huang | first1=He-liang | date=2021-03-03 | title=Emulating Quantum Teleportation of a Majorana Zero Mode Qubit| journal=Phys. Rev. Lett.| language=en| volume=126| at=090502 | doi=10.1103/PhysRevLett.126.090502| arxiv=2009.07590}}</ref>  

In 2022, researchers at the [[University of Innsbruck]] have demonstrated a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. They have performed a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code, and fault-tolerantly prepared a logical [[Magic state distillation|magic state]].<ref>{{cite journal |last1=Postler |first1=Lukas |last2=Heußen |first2=Sascha |last3=Pogorelov |first3=Ivan |last4=Rispler |first4=Manuel |last5=Feldker |first5=Thomas |last6=Meth |first6=Michael |last7=Marciniak |first7=Christian D. |last8=Stricker |first8=Roman |last9=Ringbauer |first9=Martin |last10=Blatt |first10=Rainer |last11=Schindler |first11=Philipp |last12=Müller |first12=Markus |last13=Monz |first13=Thomas |title=Demonstration of fault-tolerant universal quantum gate operations |journal=Nature |date=25 May 2022 |volume=605 |issue=7911 |pages=675–680 |doi=10.1038/s41586-022-04721-1 |pmid=35614250 |arxiv=2111.12654 |s2cid=244527180 |bibcode=2022Natur.605..675P}}</ref>

In February 2023, researchers at Google claimed to have decreased quantum errors by increasing the qubit number in experiments, they used a fault tolerant [[surface code]] measuring an error rate of 3.028% and 2.914% for a distance-3 qubit array and a distance-5 qubit array respectively.<ref>{{Cite journal |author=((Google Quantum AI)) |date=2023-02-22 |title=Suppressing quantum errors by scaling a surface code logical qubit |journal=Nature |language=en |volume=614 |issue=7949 |pages=676–681 |doi=10.1038/s41586-022-05434-1 |doi-access=free |pmid=36813892 |pmc=9946823 |bibcode=2023Natur.614..676G |issn=1476-4687}}</ref><ref>{{Cite web |last=Boerkamp |first=Martijn |date=2023-03-20 |title=Breakthrough in quantum error correction could lead to large-scale quantum computers |website=Physics World |url=https://physicsworld.com/breakthrough-in-quantum-error-correction-could-lead-to-large-scale-quantum-computers/ |access-date=2023-04-01 |language=en-GB}}</ref><ref>{{Cite web |last=Conover |first=Emily |date=2023-02-22 |title=Google's quantum computer reached an error-correcting milestone |website=ScienceNews |language=en-US |url=https://www.sciencenews.org/article/google-quantum-computer-sycamore-milestone |access-date=2023-04-01}}</ref>

In April 2024, researchers at [[Microsoft Azure Quantum|Microsoft]] claimed to have successfully tested a quantum error correction code that allowed them to achieve an error rate with logical qubits that is 800 times better than the underlying physical error rate.<ref>{{Cite web |last=Smith-Goodson |first=Paul |date=2024-04-18 |title=Microsoft And Quantinuum Improve Quantum Error Rates By 800x |website=Forbes |language=en-US |url=https://www.forbes.com/sites/moorinsights/2024/04/18/microsoft-and-quantinuum-improve-quantum-error-rates-by-800x/ |access-date=2024-07-01}}</ref>

This qubit virtualization system was used to create 4 logical qubits with 30 of the 32 qubits on Quantinuum's trapped-ion hardware. The system uses an active syndrome extraction technique to diagnose errors and correct them while calculations are underway without destroying the logical qubits.<ref>{{Cite web |last=Yirka |first=Bob |date=2024-04-05 |title=Quantinuum quantum computer using Microsoft's 'logical quantum bits' runs 14,000 experiments with no errors |website=Phys.org |language=en-US |url=https://phys.org/news/2024-04-quantinuum-quantum-microsoft-logical-bits.html |access-date=2024-07-01}}</ref>

In January 2025, researchers at [[UNSW Sydney]] managed to develop an error correction method using [[antimony]]-based materials, including [[antimonides]], leveraging high-dimensional quantum states ([[qudit]]s) with up to eight states. By encoding quantum information in the nuclear spin of a [[phosphorus]] atom embedded in [[silicon]] and employing advanced pulse control techniques, they demonstrated enhanced error resilience.<ref>{{cite journal |last=Yu |first=Xi |display-authors=et al. |year=2025 |title=Schrödinger cat states of a nuclear spin qudit in silicon |journal=Nature Physics |doi=10.1038/s41567-024-02745-0 |arxiv=2405.15494 }}</ref>